Those aren't quick filters. Those are hierarchy controls.
I'm not going to be able to do anything with just screen shots. Can you upload a workbook that demonstrates the problem? If proprietary data is an issue, hack in fake data as shown in this video:
I created the anonymous data version and checked several times along the way initially, to see if it got fixed. But then I took several additional steps to delete out the old data sourced and tabs I didn't need to display and then when I published it online the hierarchy controls were back on that table. So it wouldn't make any sense for me to attach that now because it doesn't show the problem.
So I'm going to run through those steps again and see if I an isolate exactly when the control re-appears.
In the meantime, are there any options in Tableau which configure the hierarchy controls that could also look at?
I can only guess. The first thing I would look for is something floating over the area where the +/- controls would appear. In Desktop it's only NEAR that area, but maybe publishing adjusts some positions slightly wherein the object moves just enough to block cursor access to the area where the controls would appear.